Let P be a convex polygon. A great many papers are devoted to investigation of various random values associated to the finite samples from the uniform distribution in the polygon, such as the number of the vertices of the convex hull of the sample, its circumference, probability that th convex hull has the fixed number of vertices, etc. In this note we address an inverse problem - whether and to what extent the distributions of these random values determine P. We show that Sylvester numbers, that is, the probabilities that the convex hull of m random point is a triangle for m = 4, 5, . . . determine generic polygon P unambiguously (up to affine transformations). Some general constructions, conjectures, and corollaries are presented.
All Science Journal Classification (ASJC) codes
- Applied Mathematics